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Topology of Manifolds

Code: 100114
Credits: 6
2026/2027
Degree programme Type Course
Mathematics OP 4

Contact lecturer

Name :
Natalia Castellana Vila
Email :
natalia.castellana@uab.cat

Teaching staff

Gabriel Martínez De Cestafe Pumares
Natalia Castellana Vila

Group languages

You can consult this information at the end of the document.

Prerequisites

The prerequisites are the first and second year courses as well as the Topology course. It is recommended to have taken the subject of Differential Geometry.

Objectives

In this course we introduce the most basic algebraic invariants that we can associate to a topological space (particularly to a manifold) and that provide us with a first approximation to the global properties of these objects. 
With special emphasis on the fundamental group, homology groups and cohomology algebras, as well as orientabiity.
 

Learning outcomes

  1. Effectively use bibliographies and electronic resources to obtain information.
  2. Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
  3. Develop critical thinking and reasoning and know how to communicate it effectively, both in one's own languages and in a third language.
  4. Students must be capable of collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
  5. Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  6. Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
  7. Generate innovative and competitive proposals for research and professional activities.
  8. Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
  9. Apply critical spirit and thoroughness to validate or reject both one's own arguments and those of others.
  10. Understand abstract language and in-depth demonstrations of some advanced theorems of geometry and topology.
  11. Devise demonstrations of mathematical results in the field of geometry and topology.

Contents

The course will cover the topics in the following list:


Definition and examples of homotopy of maps.

Homotopy equivalent topological spaces.

Topological manifolds. Examples.

Fundamental group and covering spaces.

Chain complexes.

Homology and the classification of connected compact surfaces.

Homology and orientability.


Furthermore, these topics will lead to the following notable results:


Cohomology.

Brouwer fixed-point theorem.

Jordan-Brouwer separation theorem.

Topological invariance of manifold dimension.

Learning activities and methodology

Title Hours ECTS Learning outcomes
Problem Sessions 15 0.6 2, 3, 4, 5, 7, 8, 9
Theory classes 30 1.2 2, 3, 4, 5, 7, 8, 9
Solving problems 30 1.2 2, 5, 10, 11
Seminars 6 0.24 2, 3, 4, 5, 7, 8, 9
Assimilations of theoretical results 45 1.8 1, 2, 5, 7
Homework 15 0.6 1, 2, 3, 5, 9, 10

In the lectures the concepts, arguments and basic results of the subject are exposed, where one expects an active participation of the students. This is complemented by problem sessions, seminars and participatory oral presentations from students, according to the contents of the course, where the students achieve the knowledge and the capacities to use these materials in readings or studies of close or more advanced topics.

 

Annotation: within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Assessment

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
Active contribution to seminar sessions 25% 4 0.16 1, 2, 3, 6, 8, 9, 10, 11
Exam 45% 4 0.16 1, 3, 4, 6, 8, 10, 11
Oral presentation, type P. 30% 1 0.04 1, 3, 4, 5, 6, 7, 8, 9, 10

The qualification is the weigthed mean of the following marks:

  • oral expositions in the seminar sessions (30%),
  • exam (40%)
  • ellaboration of a work on a proposed subject (30%)

A minimum mark of 3.5 for each evaluated activity is required. If necessary, the exam and the work will be reevaluated. The \"matrícula d'honor\" will be decided before reevaluations.

The one day assessment (avaluació unica) will take place on the same day as the final course presentations. The one day assessment will consist of the delivery of exercices (previously assignes), the final presentation and an exam,


Disclaimer: I have made my best to translate into English the Catalan version. In the unlikely case of differences between versions, we'll follow the criteria in the Catalan original version.

Bibliography

Main references:

  • W. Fulton, Algebraic topology. A first course. Graduate Texts in Mathematics, 153. Springer-Verlag, New York, 1995.
  • A. Hatcher, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. (http://www.math.cornell.edu/~hatcher/AT/ATpage.html)
  • J. M. Lee, Introduction to Topological Manifolds, , Graduate Texts in Mathematics 202, Springer-Verlag, New York, 2011.

Additional refs:

  • L. W. Tu, An introduction to manifolds. Universitext. Springer, New York, second edition, 2011.
  • R. Bott and L.W. Tu, Differential forms in algebraic topology. Graduate Texts in Mathematics, 82. Springer-Verlag, NewYork-Berlin, 1982.

Online resources:

  • Clara Löh, Algebraic Topology, https://loeh.app.uni-regensburg.de/teaching/topologie1_ws2122/lecture_notes.pdf
  • Emily Riehl, Categorical homotopy theory, https://math.jhu.edu/~eriehl/cathtpy/

Software

No specific software will be used.

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TE) Theory 1 Catalan first semester morning-mixed
(PAUL) Classroom practices 1 Catalan/Spanish first semester morning-mixed
(SEM) Seminars 1 Catalan/Spanish first semester afternoon